Physics: Momentum, Energy, Forces, and Curved Motion

In a perfectly elastic collision, momentum is conserved. The conservation of momentum is described by the equation: $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$, where $m_1$ and $m_2$ are the masses, $u_1$ and $u_2$ are the initial velocities, and $v_1$ and $v_2$ are the final velocities of the two objects.

If friction is negligible, the initial potential energy of the block at the top of the incline is equal to the final kinetic energy of the block at the bottom of the incline. This is due to the conservation of mechanical energy.

If the coefficient of kinetic friction remains constant, the acceleration of the box will remain constant as its speed increases. The acceleration is determined by the net force, which is the applied force minus the constant kinetic friction force.

The kinetic energy of a satellite in a circular orbit is given by $K = \frac{1}{2}mv^2$, where $m$ is the mass of the satellite and $v$ is its orbital speed. Using the expression for centripetal acceleration, $v^2 = \frac{GM}{r}$, where $G$ is the gravitational constant, $M$ is the mass of the Earth, and $r$ is the radius of the orbit, we can substitute for $v$ to get $K = \frac{1}{2}m\left(\frac{GM}{r}\right)$.

For a particle moving in a circular path with constant speed, the kinetic energy remains constant. However, the direction of the velocity vector is constantly changing, which means that the centripetal acceleration is also constantly changing in direction. The centripetal force, which provides the centripetal acceleration, must also change in direction continuously to keep the particle moving in a circular path.